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Theorem gsumval2 17103
 Description: Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumval2.b 𝐵 = (Base‘𝐺)
gsumval2.p + = (+g𝐺)
gsumval2.g (𝜑𝐺𝑉)
gsumval2.n (𝜑𝑁 ∈ (ℤ𝑀))
gsumval2.f (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
Assertion
Ref Expression
gsumval2 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Proof of Theorem gsumval2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval2.b . . . 4 𝐵 = (Base‘𝐺)
2 eqid 2610 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 eqid 2610 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 gsumval2.g . . . . 5 (𝜑𝐺𝑉)
65adantr 480 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
7 ovex 6577 . . . . 5 (𝑀...𝑁) ∈ V
87a1i 11 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝑀...𝑁) ∈ V)
9 gsumval2.f . . . . . . 7 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
10 ffn 5958 . . . . . . 7 (𝐹:(𝑀...𝑁)⟶𝐵𝐹 Fn (𝑀...𝑁))
119, 10syl 17 . . . . . 6 (𝜑𝐹 Fn (𝑀...𝑁))
1211adantr 480 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
13 simpr 476 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
14 df-f 5808 . . . . 5 (𝐹:(𝑀...𝑁)⟶{𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ↔ (𝐹 Fn (𝑀...𝑁) ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}))
1512, 13, 14sylanbrc 695 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶{𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
161, 2, 3, 4, 6, 8, 15gsumval1 17100 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g𝐺))
17 simpl 472 . . . . . . . . 9 (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
1817ralimi 2936 . . . . . . . 8 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦)
1918a1i 11 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦))
2019ss2rabi 3647 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦}
21 fvex 6113 . . . . . . . 8 (0g𝐺) ∈ V
2221snid 4155 . . . . . . 7 (0g𝐺) ∈ {(0g𝐺)}
23 fdm 5964 . . . . . . . . . . . . . 14 (𝐹:(𝑀...𝑁)⟶𝐵 → dom 𝐹 = (𝑀...𝑁))
249, 23syl 17 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = (𝑀...𝑁))
25 gsumval2.n . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ𝑀))
26 eluzfz1 12219 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
27 ne0i 3880 . . . . . . . . . . . . . 14 (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
2825, 26, 273syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ≠ ∅)
2924, 28eqnetrd 2849 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 ≠ ∅)
30 dm0rn0 5263 . . . . . . . . . . . . 13 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
3130necon3bii 2834 . . . . . . . . . . . 12 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
3229, 31sylib 207 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
3332adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
34 ssn0 3928 . . . . . . . . . 10 ((ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ∧ ran 𝐹 ≠ ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3513, 33, 34syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3635neneqd 2787 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅)
371, 2, 3, 4mgmidsssn0 17092 . . . . . . . . . . 11 (𝐺𝑉 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
385, 37syl 17 . . . . . . . . . 10 (𝜑 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
39 sssn 4298 . . . . . . . . . 10 ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)} ↔ ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)}))
4038, 39sylib 207 . . . . . . . . 9 (𝜑 → ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)}))
4140orcanai 950 . . . . . . . 8 ((𝜑 ∧ ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
4236, 41syldan 486 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
4322, 42syl5eleqr 2695 . . . . . 6 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
4420, 43sseldi 3566 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦})
45 oveq1 6556 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑦))
4645eqeq1d 2612 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g𝐺) + 𝑦) = 𝑦))
4746ralbidv 2969 . . . . . . 7 (𝑥 = (0g𝐺) → (∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
4847elrab 3331 . . . . . 6 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
49 oveq2 6557 . . . . . . . 8 (𝑦 = (0g𝐺) → ((0g𝐺) + 𝑦) = ((0g𝐺) + (0g𝐺)))
50 id 22 . . . . . . . 8 (𝑦 = (0g𝐺) → 𝑦 = (0g𝐺))
5149, 50eqeq12d 2625 . . . . . . 7 (𝑦 = (0g𝐺) → (((0g𝐺) + 𝑦) = 𝑦 ↔ ((0g𝐺) + (0g𝐺)) = (0g𝐺)))
5251rspcva 3280 . . . . . 6 (((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦) → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5348, 52sylbi 206 . . . . 5 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5444, 53syl 17 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5525adantr 480 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
5638ad2antrr 758 . . . . . 6 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
5715ffvelrnda 6267 . . . . . 6 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
5856, 57sseldd 3569 . . . . 5 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {(0g𝐺)})
59 elsni 4142 . . . . 5 ((𝐹𝑧) ∈ {(0g𝐺)} → (𝐹𝑧) = (0g𝐺))
6058, 59syl 17 . . . 4 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) = (0g𝐺))
6154, 55, 60seqid3 12707 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g𝐺))
6216, 61eqtr4d 2647 . 2 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
635adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
6425adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
659adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
66 simpr 476 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
671, 3, 63, 64, 65, 4, 66gsumval2a 17102 . 2 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
6862, 67pm2.61dan 828 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663  Basecbs 15695  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-0g 15925  df-gsum 15926 This theorem is referenced by:  gsumprval  17104  gsumwsubmcl  17198  gsumws1  17199  gsumccat  17201  gsumwmhm  17205  gsumval3  18131  gsummptfzcl  18191  gsumncl  29941  gsumnunsn  29942
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